Exponential Concentration for Zeroes of Stationary Gaussian Processes

We show that for any centered stationary Gaussian process of absolutely integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $\eta T$ of its mean value, up to an exponentially small in $T$ probability.

On the Probability That a Stationary Gaussian Process With Spectral Gap Remains Non-negative on a Long Interval

Let $f$ be a zero mean continuous stationary Gaussian process on $\mathbb{R}$ whose spectral measure vanishes in a $\delta $-neighborhood of the origin. Then, the probability that $f$ stays non-negative on an interval of length $L$ is at most $e^{-c\delta ^2 L^2}$ with some absolute $c>0$ and the result is sharp without additional assumptions.

From random matrices to long range dependence

Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process. This is similar to the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues after a different scaling. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.